Let $x\geq 0$ be a positive element in a von Neumann algebra $\mathcal M.$ Then b y functional calculus the projection $e_\lambda=1_{[0.\lambda)}(x)$ has the property that $e_\lambda$ commutes with $x$ and $e_\lambda x e_\lambda<\lambda.$ Does this characterize $e_\lambda$? That is if $e$ is a projection which commutes with $x$ and $exe<\lambda$ , do we have that $e=e_\lambda$?

Let $x\geq 0$ be a positive element in a von Neumann algebra $\mathcal M.$ Then b y functional calculus the projection $e_\lambda=1_{[0.\lambda)}(x)$ has the property that $e_\lambda$ commutes with $x$ and $e_\lambda x e_\lambda<\lambda.$ Does this characterize $e_\lambda$? That is if $e$ is the largest projection which commutes with $x$ and $exe<\lambda$ , do we have that $e=e_\lambda$?